It is known that lipid membranes become permeable in their melting regime. In conductance measurements on black lipid membranes one finds that conduction takes place via quantized events closely resembling those reported for protein ion channels. Here, we present data of ion currents through black lipid membranes in the presence and absence of the anesthetics octanol and ethanol, and compare them to a statistical thermodynamics model using parameters that are obtained from experimental calorimetric data. The conductance steps in pure lipid membrane suggest aqueous pores with the size of approximately one lipid cross-section. We model the permeability by assuming empty lattice sites of the size of one lipid. We find that pore formation in the melting transition regime is facilitated by the increase of the lateral compressibility that expresses itself in the area fluctuations, a process that is related to critical opalescence in two dimensions. Anesthetics alter the permeability by affecting the thermodynamic state of the membrane and by shifting the heat capacity profiles.